The multiple-quantum NMR spectroscopy has an extensive application in determination of the bio-macro-molecular structures and in the investigation of the properties of a variety of physical materials. In quantum computation the multiple-quantum transition processes have been used to construct the quantum circuits, quantum algorithms, and quantum simulations. The multiple-quantum operator algebra spaces are closely related to the symmetries of a multiple-spin quantum system. They may have an important effect on the multiple-quantum transition processes and the multiple-quantum NMR spectroscopy of the spin system. Here gives a brief introduction to the multiple-quantum operator algebra spaces.
We define quantum phase in terms of inverses of annihilation and creation operators. We show that like Susskind - Glogower phase operators, the measured phase operators and the unitary phase operators can be defined in terms of the inverse operators. However, for the unitary phase operator the Hilbert space includes the negative energy states. The quantum phase in inverse operator representation may find the applications in the field of quantum optics particularly in the squeezed states.
We present a robust method for quantum process tomography, which yields a set of Lindblad operators that optimally fit the measured density operators at a sequence of time points. The benefits of this method are illustrated using a set of liquid-state Nuclear Magnetic Resonance (NMR) measurements on a molecule containing two coupled hydrogen nuclei, which are sufficient to fully determine its relaxation superoperator. It was found that the complete positivity constraint, which is necessary for the existence of the Lindblad operators, was also essential for obtaining a robust fit to the measure…
Operator quantum error-correction is a technique for robustly storing quantum information in the presence of noise. It generalizes the standard theory of quantum error-correction, and provides a unified framework for topics such as quantum error-correction, decoherence-free subspaces, and noiseless subsystems. This paper develops (a) easily applied algebraic and information-theoretic conditions which characterize when operator quantum error-correction is feasible; (b) a representation theorem for a class of noise processes which can be corrected using operator quantum error-correction; and (c)…
A Hermitian quantum phase operator is formulated that mirrors the classical phase variable with proper time dependence and satisfies trigonometric identities. The eigenstates of the phase operator are solved in terms of Gegenbauer ultraspherical polynomials in the number state representation.
Explicit expressions for most interesting quantum operators in optical tomography representation are found. General formalism of symbols of operators is presented in optical tomographic representation. The symbols of the operators are found explicitly for physical quantities.
We show that the quantum mechanical momentum and angular momentum operators are fixed by the Noether theorem for the classical Hamiltonian field theory we proposed.
By using the formalism of photon creation operators, we present the simplest description of the effect of quantum teleportation and describe its closest classical analog.
The attempt to equate operator quantum error correction (quant-ph/0504189v1) with the quantum computer condition (quant-ph/0507141) in version two of quant-ph/0504189 is shown to be invalid.
The true level crossing in the asymmetric quantum Rabi model without any obvious symmetry can be exhibited in the energy spectrum if the qubit bias is a multiple of the cavity frequency, which should imply the existence of the hidden symmetry. In this work, within a Bogoliubov operator approach, we can readily derive the symmetry operators associated with the hidden symmetry hierarchically for arbitrary multiples. The symmetry operators for small multiples in the literature can be extremely easily reproduced in our general scheme. In addition, a general parity operator is defined through the s…
The time evolution of some quantum states can be slowed down or even stopped under frequent measurements. This is the usual quantum Zeno effect. Here, we report an operator quantum Zeno effect, in which the evolution of some physical observables is slowed down through measurements even though the quantum state changes randomly with time. Based on the operator quantum Zeno effect, we show how we can protect quantum information from decoherence with two-qubit measurements, realizable with noisy two-qubit interactions.
We study the entanglement of unitary operators on $d_1\times d_2$ quantum systems. This quantity is closely related to the entangling power of the associated quantum evolutions. The entanglement of a class of unitary operators is quantified by the concept of concurrence.
Aim of this paper is to show the possible significance, and usefulness, of various non-selfadjoint operators for suitable Observables in non relativistic and relativistic quantum mechanics, and in quantum electrodynamics. More specifically, this work starts dealing with: (i) the maximal hermitian (but not selfadjoint) Time operator in non-relativistic quantum mechanics and in quantum electrodynamics; and with: (ii) the problem of the four-position and four-momentum operators, each one with its hermitian and anti-hermitian parts, for relativistic spin-zero particles. Afterwards, other physicall…
In the framework of any quantum theory in the Schroedinger picture a general operator time concept is given. For this purpose certain systems are emphasized as ideal quantum clocks. Their definition follows heuristically from a common property of ideal clocks and from general postulates of traditional quantum theory. Any such ideal quantum clock allows the definition of a symmetric time operator T. T and the Hamiltonian H necessarily satisfy the time-energy uncertainty relation. The argument of Pauli against the existence of any time operator does not strike, because T is symmetric but not sel…
Pseudo-random operators consist of sets of operators that exhibit many of the important statistical features of uniformly distributed random operators. Such pseudo-random sets of operators are most useful whey they may be parameterized and generated on a quantum processor in a way that requires exponentially fewer resources than direct implementation of the uniformly random set. Efficient pseudo-random operators can overcome the exponential cost of random operators required for quantum communication tasks such as super-dense coding of quantum states and approximately secure quantum data-hiding…
We present a theoretical result, which is based on the linear algebra theory (similar operators). The obtained theoretical results optimize the experimental technique to construct quantum computer e.g., reduces the number of steps to perform the logical CNOT (XOR) operation. The present theoretical technique can also be generalized to the other operators in in quantum computation and information theory.
The components of a quantum computer are quantum subsystems which have a complex internal structure. This structure is determined by short-range interactions which are appropriately described in terms of local gauge fields of the first kind. Any modification of their state would produce, in general, a new type of internal error, called local error, in the quantum state of the computer. We suggest that the general treatement of the local errors produced by a gauge multiplet can be done in the framework of the Algebraic Quantum Field Theory. A recovery operator is constructed from the first prin…
We propose a quantum expected value theory for decision-making under uncertainty. Quantum density operator as value operator is proposed to simulate people's subjective beliefs. Value operator guides people to choose corresponding actions based on their subjective beliefs through objective world. The value operator can be constructed from quantum gates and logic operations as a quantum decision tree. The genetic programming is used to optimize and auto-generate quantum decision trees.
Quantum machine learning for the quantum lattice Boltzmann method: Trainability of variational quantum circuits for the nonlinear collision operator across multiple time steps
A relativistic spin operator is to be the difference between the total and orbital angular momentum. As the unique position operator for a localized state, the remarkable Newton-Wigner position operator, which has all desirable commutation relations as a position operator, can give a proper spin operator. Historically important three spin operators respectively proposed by Bogolubov et al., Pryce, and Foldy-Woutheysen are investigated to manifest a corresponding spin operator to the Newton-Wigner position operator. We clarify a unique spin operator in relativistic quantum mechanics described b…
Although quantum logic by using exogenous approach has been proposed for reasoning about closed quantum systems, an improvement would be worth to study quantum logic based on density operators instead of unit vectors in the state logic point of view. In order to achieve this, we build an exogenous quantum operator logic(EQOL) based on density operators for reasoning about open quantum systems. We show that this logic is sound and complete. Just as the exogenous quantum propositional logic(EQPL), by applying exogenous approach, EQOL is extended from the classical propositional logic, and is use…
We describe the construction of quantum gates (unitary operators) from boolean functions and give a number of applications. Both non-reversible and reversible boolean functions are considered. The construction of the Hamilton operator for a quantum gate is also described with the Hamilton operator expressed as spin system. Computer algebra implementations are provided.
After a brief introduction recalling how, in the limit in which the mass and the electric charge of the electron and the positron tend to zero, Quantum Electrodynamics reduces to a collection of uncoupled quantum supersymmetric harmonic oscillators, the phase operator of the quantum fermionic harmonic oscillator and of the quantum supersymmetric harmonic oscillator are introduced and their properties analyzed. It is then shown that the phase operator of a supersymmetric harmonic oscillator is a Goldstone operator at any strictly positive temperature (finite or infinite).
We establish operator structure identities for quantum channels and their error-correcting and private codes, emphasizing the complementarity relationship between the two perspectives. Relevant structures include correctable and private operator algebras, and operator spaces such as multiplicative domains and nullspaces of quantum channels and their complementary maps. For the case of privatizing to quantum states, we also derive related dimension inequalities that may be viewed as operator algebra uncertainty relations.
We conduct the first detailed analysis in quantum information of recently derived operator relations from the study of quantum one-way local operations and classical communications (LOCC). We show how operator structures such as operator systems, operator algebras, and Hilbert C*-modules all naturally arise in this setting, and we make use of these structures to derive new results and new derivations of some established results in the study of LOCC. We also show that perfect distinguishability under one-way LOCC and under arbitrary operations is equivalent for several families of operators tha…
We introduce unitary quantum phase operators for material particles. We carry out a model study on quantum phases of interacting bosons in a symmetric double-well potential in terms of unitary and commonly-used non-unitary phase operators and compare the results for different number of bosons. We find that the results for unitary quantum phase operators are significantly different from those for non-unitary ones especially in the case of low number of bosons. We introduce unitary operators corresponding to the quantum phase-difference between two single-particle states of fermions. As an appli…
Subspace-selective unitary manipulation based on the Hilbert-space symmetric structures in the multiple-quantum operator algebra spaces in the quantum-computing speedup theory
A consistent realization of the quantum operators corresponding to the canonically conjugate phase and number variables is proposed, resorting to the irreducible unitary representations of the Lie algebra su(1,1), as proposed by Kastrup.
Tensor product operators on finite dimensional Hilbert spaces are studied. The focus is on bilinear tensor product operators. A tensor product operator on a pair of Hilbert spaces is a maximally general bilinear operator into a target Hilbert space. By 'maximally general' is meant every bilinear operator from the same pair of spaces to any Hilbert space factors into the composition of the tensor product operator with a uniquely determined linear mapping on the target space. There are multiple distinct tensor product operators of the same type; there is no "the" tensor product. Distinctly diffe…
Electron transport in realistic physical and chemical systems often involves the non-trivial exchange of energy with a large environment, requiring the definition and treatment of open quantum systems. Because the time evolution of an open quantum system employs a non-unitary operator, the simulation of open quantum systems presents a challenge for universal quantum computers constructed from only unitary operators or gates. Here we present a general algorithm for implementing the action of any non-unitary operator on an arbitrary state on a quantum device. We show that any quantum operator ca…
In a certain sense we generalize the recently introduced and extensively studied notion called quantum Rényi divergence (in another name, sandwiched Rényi relative entropy) and describe the structures of corresponding symmetries. More precisely, we characterize all transformations on the set of density operators which leave our new general quantity invariant and also determine the structure of all bijective transformations on the cone of positive definite operators which preserve the quantum Rényi divergence.
We study operator entanglement of the quantum chaotic evolutions. This study shows that properties of the operator entanglement production are qualitatively similar to the properties reported in literature about the pure state entanglement production. This similarity establishes that the operator entanglement quantifies {\it intrinsic} entangling power of an operator. The term `intrinsic' suggests that this measure is independent of any specific choice of initial states.
Operator scrambling is a crucial ingredient of quantum chaos. Specifically, in the quantum chaotic system, a simple operator can become increasingly complicated under unitary time evolution. This can be diagnosed by various measures such as square of the commutator (out-of-time-ordered correlator), operator entanglement entropy etc. In this paper, we discuss operator scrambling in three representative models: a chaotic spin-$1/2$ chain with spatially local interactions, a 2-local spin model and the quantum linear map. In the first two examples, although the speeds of scrambling are quite diffe…
The anti self-adjoint operators of imaginary coordinate and momentum, together with the self-adjoint operators of real coordinate, momentum, energy and time are used in construction of the quantum field theory in operator form. This formalism, being free of many dubious mathematical situations characteristic to the standard treatment of quantum fields, is applied in formulation of the new gauge condition for electromagnetic field that describes the absence of the scalar and longitudinal photons. Proposed formalism offers adequate theoretical framework for proper description of the multicompone…
The spectral analysis of a non-Hermitian unbounded operator appearing in quantum physics is our main concern. The properties of such an operator are essentially different from those of Hermitian Hamiltonians, namely due to spectral instabilities. We demonstrate that the considered operator and its adjoint can be diagonalized when expressed in terms of certain conveniently constructed operators. We show that their eigenfunctions constitute complete systems, but do not form Riesz bases. Attempts to overcome this difficulty in the quantum mechanical set up are pointed out.
We introduce a model of the quantum Brownian motion coupled to a classical neat bath by using the operator differential proposed in the quantum analysis. We then define the heat operator by adapting the idea of the stochastic energetics. The introduced operator satisfies the relations which are analogous to the first and second laws of thermodynamics.
The associators/antiassociators for the product of four non-associative operators are deduced. By analogy with SU(3) gauge theory the notion of colorless (white) operators is introduced. Some properties of white operators are considered. It is hypothesized that white operators do not give any contribution to corresponding associators/antiassociators. It is suggested that the observables in a non-associative quantum theory correspond to the white operators only.
The interrelations between the two definitions of momentum operator, via the canonical energy-momentum tensorial operator and as translation operator (on the operator space), are studied in quantum field theory. These definitions give rise to similar but, generally, different momentum operators, each of them having its own place in the theory. Some speculations on the relations between quantum field theory and quantum mechanics are presented.
We propose a variational method for identifying lattice operators in a critical quantum spin chain with scaling operators in the underlying conformal field theory (CFT). In particular, this allows us to build a lattice version of the primary operators of the CFT, from which we can numerically estimate the operator product expansion coefficients $C_{αβγ}^{\textrm{ CFT}}$. We demonstrate the approach with the critical Ising quantum spin chain.
Given a completely integrable system, we associate to any connection on its invariant tori fibred over a parameter manifold the classical and quantum holonomy operator (generalized Berry's phase factor), without any adiabatic approximation.
